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Sep 22, 2013
09/13

by
C. Chandre; R. S. MacKay

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We construct an approximate renormalization transformation for Hamiltonian systems with three degrees of freedom in order to study the break-up of invariant tori with three incommensurate frequencies which belong to the cubic field $Q(\tau)$, where $\tau^3+\tau^2-2\tau-1=0$. This renormalization has two fixed points~: a stable one and a hyperbolic one with a codimension one stable manifold. We compute the associated critical exponents that characterize the universality class for the break-up of...

Source: http://arxiv.org/abs/nlin/0001033v1

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Sep 24, 2013
09/13

by
C. Chandre

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We analyze by a renormalization method, the dynamics of a particle in a infinite square-well potential driven by an external monochromatic field. This method set up for Hamiltonian systems with two degrees of freedom allows us to analyze precisely the stability of the trajectories of the particle as a function of the amplitude $\epsilon$ of the external field. We compute numerical values of $\epsilon$ for which the motion of the particle with frequency $\omega$ is broken and a transition to a...

Source: http://arxiv.org/abs/nlin/0103059v1

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20

Jun 26, 2018
06/18

by
M Perin; C Chandre; P. J. Morrison; E Tassi

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Fluid reductions of the Vlasov-Amp{\`e}re equations that preserve the Hamiltonian structure of the parent kinetic model are investigated. Hamiltonian closures using the first four moments of the Vlasov distribution are obtained, and all closures provided by a dimensional analysis procedure for satisfying the Jacobi identity are identified. Two Hamiltonian models emerge, for which the explicit closures are given, along with their Poisson brackets and Casimir invariants.

Topics: Nonlinear Sciences, Chaotic Dynamics, Physics, Plasma Physics, Fluid Dynamics

Source: http://arxiv.org/abs/1502.04639

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Sep 22, 2013
09/13

by
C. Chandre; H. R. Jauslin; G. Benfatto

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We construct an approximate renormalization scheme for Hamiltonian systems with two degrees of freedom. This scheme is a combination of Kolmogorov-Arnold-Moser (KAM) theory and renormalization-group techniques. It makes the connection between the approximate renormalization procedure derived by Escande and Doveil, and a systematic expansion of the transformation. In particular, we show that the two main approximations, consisting in keeping only the quadratic terms in the actions and the two...

Source: http://arxiv.org/abs/chao-dyn/9805019v1

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3.0

Jun 30, 2018
06/18

by
C. Chandre

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First-class constraints constitute a potential obstacle to the computation of a Poisson bracket in Dirac's theory of constrained Hamiltonian systems. Using the pseudoinverse instead of the inverse of the matrix defined by the Poisson brackets between the constraints, we show that a Dirac-Poisson bracket can be constructed, even if it corresponds to an incomplete reduction of the original Hamiltonian system. The uniqueness of Dirac brackets is discussed.

Topics: Nonlinear Sciences, Chaotic Dynamics

Source: http://arxiv.org/abs/1412.4922

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2.0

Jun 30, 2018
06/18

by
Michael Norman; C. Chandre; T. Uzer; Peijie Wang

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We revisit the stabilization of ionization of atoms subjected to a superintense laser pulse using nonlinear dynamics. We provide an explanation for the lack of complete ionization at high intensity and for the decrease of the ionization probability as intensity is increased. We investigate the role of each part of the laser pulse (ramp-up, plateau, ramp-down) in this process. We emphasize the role of the choice for the ionization criterion, energy versus distance criterion.

Topics: Physics, Nonlinear Sciences, Atomic Physics, Chaotic Dynamics

Source: http://arxiv.org/abs/1412.2205

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Sep 18, 2013
09/13

by
C. Chandre; J. Laskar; G. Benfatto; H. R. Jauslin

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We consider a class of Hamiltonians with three degrees of freedom that can be mapped into quasi-periodically driven pendulums. The purpose of this paper is to determine the threshold of the break-up of invariant tori with a specific frequency vector. We apply two techniques: the frequency map analysis and renormalization-group methods. The renormalization transformation acting on a Hamiltonian is a canonical change of coordinates which is a combination of a partial elimination of the irrelevant...

Source: http://arxiv.org/abs/nlin/0105027v1

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Sep 20, 2013
09/13

by
S. Huang; C. Chandre; T. Uzer

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We discuss the influence of periodic orbits on the dissociation of a model diatomic molecule driven by a strong bichromatic laser fields. Through the stability of periodic orbits we analyze the dissociation probability when parameters like the two amplitudes and the phase lag between the laser fields, are varied. We find that qualitative features of dissociation can be reproduced by considering a small set of short periodic orbits. The good agreement with direct simulations demonstrates the...

Source: http://arxiv.org/abs/0802.3834v2

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Sep 20, 2013
09/13

by
S. Huang; C. Chandre; T. Uzer

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The multiphoton ionization of hydrogen by a strong bichromatic microwave field is a complex process prototypical for atomic control research. Periodic orbit analysis captures this complexity: Through the stability of periodic orbits we can match qualitatively the variation of experimental ionization rates with a control parameter, the relative phase between the two modes of the field. Moreover, an empirical formula reproduces quantum simulations to a high degree of accuracy. This quantitative...

Source: http://arxiv.org/abs/nlin/0612056v1

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Sep 18, 2013
09/13

by
R. Chabreyrie; D. Vainchtein; C. Chandre; P. Singh; N. Aubry

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The design of strategies to generate efficient mixing is crucial for a variety of applications, particularly digital microfluidic devices that use small "discrete" fluid volumes (droplets) as fluid carriers and microreactors. In recent work, we have presented an approach for the generation and control of mixing inside a translating spherical droplet. This was accomplished by considering Stokes' flow within a droplet proceeding downstream to which we have superimposed time dependent...

Source: http://arxiv.org/abs/0809.0455v1

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Sep 20, 2013
09/13

by
C. Chandre; M. Govin; H. R. Jauslin

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We analyze the breakup of invariant tori in Hamiltonian systems with two degrees of freedom using a combination of KAM theory and renormalization-group techniques. We consider a class of Hamiltonians quadratic in the action variables that is invariant under the chosen KAM transformations, following the approach of Thirring. The numerical implementation of the transformation shows that the KAM iteration converges up to the critical coupling at which the torus breaks up. By combining this...

Source: http://arxiv.org/abs/chao-dyn/9802022v1

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Sep 22, 2013
09/13

by
R. Paškauskas; C. Chandre; T. Uzer

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Finding the causes for the nonstatistical vibrational energy relaxation in the planar carbonyl sulfide (OCS) molecule is a longstanding problem in chemical physics: Not only is the relaxation incomplete long past the predicted statistical relaxation time, but it also consists of a sequence of abrupt transitions between long-lived regions of localized energy modes. We report on the phase space bottlenecks responsible for this slow and uneven vibrational energy flow in this Hamiltonian system...

Source: http://arxiv.org/abs/0811.4563v1

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4.0

Jun 27, 2018
06/18

by
S. A. Berman; C. Chandre; T. Uzer

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We find that Coulomb focusing persists even when the Coulomb field is barely noticeable compared with the laser field. Delayed recollisions proliferate in this regime and bring back energy slightly above the 3.17 U_p high-harmonic cutoff, in stark contradiction with the Strong Field Approximation. We investigate the nonlinear-dynamical phase space structures which underlie this dynamics. It is found that the energetic delayed recollisions are organized by a reduced number of periodic orbits and...

Topics: Chaotic Dynamics, Atomic Physics, Physics, Nonlinear Sciences

Source: http://arxiv.org/abs/1506.00805

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Jul 22, 2013
07/13

by
R. Bachelard; C. Chandre; D. Fanelli; X. Leoncini; M. Vittot

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The intensity of an electromagnetic wave interacting self-consistently with a beam of charged particles, as in a Free Electron Laser, displays large oscillations due to an aggregate of particles, called the macro-particle. In this article, we propose a strategy to stabilize the intensity by destabilizing the macro-particle. This strategy involves the study of the linear stability of a specific periodic orbit of a mean-field model. As a control parameter - the amplitude of an external wave - is...

Source: http://arxiv.org/abs/0807.0286v1

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Sep 18, 2013
09/13

by
C. Chandre; H. R. Jauslin; G. Benfatto; A. Celletti

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We construct an approximate renormalization transformation that combines Kolmogorov-Arnold-Moser (KAM)and renormalization-group techniques, to analyze instabilities in Hamiltonian systems with three degrees of freedom. This scheme is implemented both for isoenergetically nondegenerate and for degenerate Hamiltonians. For the spiral mean frequency vector, we find numerically that the iterations of the transformation on nondegenerate Hamiltonians tend to degenerate ones on the critical surface....

Source: http://arxiv.org/abs/chao-dyn/9903018v1

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Sep 18, 2013
09/13

by
C. Chandre; P. Moussa

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We construct an approximate renormalization for Hamiltonian systems with two degrees of freedom in order to study the break-up of invariant tori with arbitrary frequency. We derive the equation of the critical surface of the renormalization map, and we compute the scaling behavior of the critical function of one-parameter families of Hamiltonians, near rational frequencies. For the forced pendulum model, we find the same scaling law found for the standard map in [Carletti and Laskar, preprint...

Source: http://arxiv.org/abs/nlin/0005020v1

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35

Sep 20, 2013
09/13

by
C. Chandre; M. Govin; H. R. Jauslin; H. Koch

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In this article, we describe a new renormalization-group scheme for analyzing the breakup of invariant tori for Hamiltonian systems with two degrees of freedom. The transformation, which acts on Hamiltonians that are quadratic in the action variables, combines a rescaling of phase space and a partial elimination of irrelevant (non-resonant) frequencies. It is implemented numerically for the case applying to golden invariant tori. We find a nontrivial fixed point and compute the corresponding...

Source: http://arxiv.org/abs/chao-dyn/9802023v1

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51

Sep 18, 2013
09/13

by
C. Chandre

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We consider here two discrete versions of the modified KdV equation. In one case, some solitary wave solutions, B\"acklund transformations and integrals of motion are known. In the other one, only solitary wave solutions were given, and we supply the corresponding results for this equation. We also derive the integrability of the second equation and give a transformation which links the two models.

Source: http://arxiv.org/abs/solv-int/9809007v1

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Sep 19, 2013
09/13

by
S. Huang; C. Chandre; T. Uzer

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We present a control procedure to reduce the stochastic ionization of hydrogen atom in a strong microwave field by adding to the original Hamiltonian a comparatively small control term which might consist of an additional set of microwave fields. This modification restores select invariant tori in the dynamics and prevents ionization. We demonstrate the procedure on the one-dimensional model of microwave ionization.

Source: http://arxiv.org/abs/nlin/0610067v1

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Sep 22, 2013
09/13

by
C. Chandre; H. R. Jauslin

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We study an approximate renormalization-group transformation to analyze the breakup of invariant tori for three degrees of freedom Hamiltonian systems. The scheme is implemented for the spiral mean torus. We find numerically that the critical surface is the stable manifold of a critical nonperiodic attractor. We compute scaling exponents associated with this fixed set, and find that they can be expected to be universal.

Source: http://arxiv.org/abs/chao-dyn/9807007v2

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Sep 19, 2013
09/13

by
R. Paškauskas; C. Chandre; T. Uzer

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Vibrational energy flows unevenly in molecules, repeatedly going back and forth between trapping and roaming. We identify bottlenecks between diffusive and chaotic behavior, and describe generic mechanisms of these transitions, taking the carbonyl sulphide molecule OCS as a case study. The bottlenecks are found to be lower-dimensional tori; their bifurcations and unstable manifolds govern the transition mechanisms.

Source: http://arxiv.org/abs/0708.1893v1

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Sep 21, 2013
09/13

by
C. Chandre

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We consider the break-up of invariant tori in Hamiltonian systems with two degrees of freedom with a frequency which belongs to a cubic field. We define and construct renormalization-group transformations in order to determine the threshold of the break-up of these tori. A first transformation is defined from the continued fraction expansion of the frequency, and a second one is defined with a fixed frequency vector in a space of Hamiltonians with three degrees of freedom.

Source: http://arxiv.org/abs/nlin/0203063v1

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Sep 21, 2013
09/13

by
G. Ciraolo; C. Chandre; R. Lima; M. Vittot; M. Pettini

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We present a technique to control chaos in Hamiltonian systems which are close to integrable. By adding a small and simple control term to the perturbation, the system becomes more regular than the original one. We apply this technique to a forced pendulum model and show numerically that the control is able to drastically reduced chaos.

Source: http://arxiv.org/abs/nlin/0311009v1

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Sep 20, 2013
09/13

by
C. Chandre; David Farrelly; T. Uzer

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We analyze the classical phase space of the hydrogen atom in crossed magnetic and circularly polarized microwave fields in the high frequency regime, using the Chirikov resonance overlap criterion and the renormalization map. These methods are used to compute thresholds to large scale chaos and to ionization. The effect of the magnetic field is a strong stabilization of a set of invariant tori which bound the trajectories and prevent stochastic ionization. In order to ionize, larger amplitudes...

Source: http://arxiv.org/abs/nlin/0204033v1

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51

Sep 18, 2013
09/13

by
R. Chabreyrie; C. Chandre; P. Singh; N. Aubry

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The ability to generate complete, or almost complete, chaotic mixing is of great interest in numerous applications, particularly for microfluidics. For this purpose, we propose a strategy that allows us to quickly target the parameter values at which complete mixing occurs. The technique is applied to a time periodic, two-dimensional electro-osmotic flow with spatially and temporally varying Helmoltz-Smoluchowski slip boundary conditions. The strategy consists of following the linear stability...

Source: http://arxiv.org/abs/1102.1448v1

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Sep 20, 2013
09/13

by
C. Chandre; H. R. Jauslin

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We review a formulation of a renormalization-group scheme for Hamiltonian systems with two degrees of freedom. We discuss the renormalization flow on the basis of the continued fraction expansion of the frequency. The goal of this approach is to understand universal scaling behavior of critical invariant tori.

Source: http://arxiv.org/abs/chao-dyn/9810009v1

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Jul 20, 2013
07/13

by
C. Chandre; H. R. Jauslin

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We analyze the stability of invariant tori for Hamiltonian systems with two degrees of freedom by constructing a transformation that combines Kolmogorov-Arnold-Moser theory and renormalization-group techniques. This transformation is based on the continued fraction expansion of the frequency of the torus. We apply this transformation numerically for arbitrary frequencies that contain bounded entries in the continued fraction expansion. We give a global picture of renormalization flow for the...

Source: http://arxiv.org/abs/chao-dyn/9906002v1

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Sep 21, 2013
09/13

by
J. Squire; H. Qin; W. M. Tang; C. Chandre

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We present a new variational principle for the gyrokinetic system, similar to the Maxwell-Vlasov action presented in Ref. 1. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid velocity and particle distribution function. Using a Legendre transform, we explicitly derive the field theoretic Hamiltonian structure of the system. This is carried out with a modified Dirac theory of constraints, which is used to construct meaningful brackets...

Source: http://arxiv.org/abs/1301.6066v2

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Sep 19, 2013
09/13

by
S. Huang; C. Chandre; T. Uzer

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We investigate the multiphoton ionization of hydrogen driven by a strong bichromatic microwave field. In a regime where classical and quantum simulations agree, periodic orbit analysis captures the mechanism: Through the linear stability of periodic orbits we match qualitatively the variation of experimental ionization rates with control parameters such as the amplitudes of the two modes of the field or their relative phases. Moreover, we discuss an empirical formula which reproduces quantum...

Source: http://arxiv.org/abs/0708.1292v2

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Sep 19, 2013
09/13

by
C. Chandre; S. Wiggins; T. Uzer

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We describe a method for analyzing the phase space structures of Hamiltonian systems. This method is based on a time-frequency decomposition of a trajectory using wavelets. The ridges of the time-frequency landscape of a trajectory, also called instantaneous frequencies, enable us to analyze the phase space structures. In particular, this method detects resonance trappings and transitions and allows a characterization of the notion of weak and strong chaos. We illustrate the method with the...

Source: http://arxiv.org/abs/nlin/0209015v1

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Sep 20, 2013
09/13

by
M. Govin; C. Chandre; H. R. Jauslin

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We study the stability and breakup of invariant tori in Hamiltonian flows using a combination of Kolmogorov-Arnold-Moser (KAM) theory and renormalization-group techniques. We implement the scheme numerically for a family of Hamiltonians quadratic in the actions to analyze the strong coupling regime. We show that the KAM iteration converges up to the critical coupling at which the torus breaks up. Adding a renormalization consisting of a rescaling of phase space and a shift of resonances allows...

Source: http://arxiv.org/abs/chao-dyn/9802021v1

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Sep 23, 2013
09/13

by
G. Ciraolo; F. Briolle; C. Chandre; E. Floriani; R. Lima; M. Vittot; M. Pettini; C. Figarella; P. Ghendrih

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It is shown that a relevant control of Hamiltonian chaos is possible through suitable small perturbations whose form can be explicitly computed. In particular, it is possible to control (reduce) the chaotic diffusion in the phase space of a Hamiltonian system with 1.5 degrees of freedom which models the diffusion of charged test particles in a turbulent electric field across the confining magnetic field in controlled thermonuclear fusion devices. Though still far from practical applications,...

Source: http://arxiv.org/abs/nlin/0312037v1

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Sep 18, 2013
09/13

by
R. Chabreyrie; D. Vainchtein; C. Chandre; P. Singh; N. Aubry

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The use of microscopic discrete fluid volumes (i.e., droplets) as microreactors for digital microfluidic applications often requires mixing enhancement and control within droplets. In this work, we consider a translating spherical liquid droplet to which we impose a time periodic rigid-body rotation which we model using the superposition of a Hill vortex and an unsteady rigid body rotation. This perturbation in the form of a rotation not only creates a three-dimensional chaotic mixing region,...

Source: http://arxiv.org/abs/0809.3728v1

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Sep 18, 2013
09/13

by
C. Chandre; H. R. Jauslin

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We give a proof of the KAM theorem on the existence of invariant tori for weakly perturbed Hamiltonian systems, based on Thirring's approach for Hamiltonians that are quadratic in the action variables. The main point of this approach is that the iteration of canonical transformations on which the proof is based stays within the space of quadratic Hamiltonians. We show that Thirring's proof for nondegenerate Hamiltonians can be adapted to Hamiltonians with degenerate twist. This case, in fact,...

Source: http://arxiv.org/abs/chao-dyn/9803005v2

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Sep 21, 2013
09/13

by
C. Chandre; T. Uzer

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We find that chaos in the stochastic ionization problem develops through the break-up of a sequence of noble tori. In addition to being very accurate, our method of choice, the renormalization map, is ideally suited for analyzing properties at criticality. Our computations of chaos thresholds agree closely with the widely used empirical Chirikov criterion.

Source: http://arxiv.org/abs/nlin/0203062v1